The curved uncut chip thickness model: A general geometric model for mechanistic cutting force predictions

Cutting force prediction plays an important role in modern manufacturing systems to effectively design cutters, fixtures, and machine tools. A novel mechanics model of parametric helical-end mills is systematically presented for three-dimensional (3D) cutting force prediction in the article, which is different from mechanistic approach and Oxley’s predictive machining theory in model formulation and shear stress identification process. The single-flute cutting edge and multiflute cutting edge of helical-end mills are modelled according to kinematic analysis with vector algebra. Based on Merchant’s oblique cutting theory, a new mechanics model of 3D cutting force with runout has been developed. Meanwhile, the asynchronous problem between predicted and measured curves is solved by adjusting phase angle to minimize the average deviation. After minimizing the asynchronous phase angle deviation, shear stress can be estimated directly using corresponding peak-to-peak ratio or valley-to-valley ratio of the predicted curves and the measured curves in X- and Y-directions of an arbitrary selected milling test. To assess the feasibility of the general model, over 100 milling experiments of aluminium alloy (7075) using flat-end mills and ball-end mills were conducted, respectively, and numerical tests implemented in time domain on MathWorks platform. The comparative results indicated that the predicted and the measured waveforms were quite satisfied in both pulsation pattern and period.

In this paper, a novel tool wear mechanism in low-frequency vibration–assisted drilling (LFVAD) of carbon fiber–reinforced plastic (CFRP)/Ti stacks and its individual layer has been proposed. Firstly, the difference of the effective rake angle, effective clearance angle, undeformed chip thickness, and tool-workpiece contact length between low-frequency vibration–assisted drilling and conventional drilling (CD) was analyzed based on the kinematic model. Then, the strength at different positions of the cutting edge was evaluated. Furthermore, the drilling experiments of CFRP/Ti stacks, CFRP plate, and Ti plate individually were taken under LFVAD and CD processes. Finally, the tool wear mechanism and the benefit of CFRP/Ti stacks and CFRP plate processed by LFVAD were expounded by kinematic analysis results, the analysis of drilling force, SEM images, and the profile of the cutting edge. The results show that the inner edge close to the chisel edge could be the weakest position of the drill in LFVAD. As the mean drilling force of LFVAD is lower than CD, the total wear volume of LFVAD is smaller than CD. Due to the change in the resultant cutting speed angle and the undeformed chip thickness, LFVAD produces the longer tool-workpiece contact length on the rake face and the flank face, which directly leads to a more uniform tool wear and a self-sharpening effect on the cutting edge. Thus, the reduction in the effective rake of the drill used in LFVAD is obviously smaller than the reduction in conventional drilling. The reduction in the effective clearance angle of the drill used in the LFVAD is also less than the reduction in conventional drilling, while this difference is attenuated as the cutting radius increases. The edge radius of the cutting edge is found to be much smaller than CD. In summary, the difference in the cutting edge geometry between LFVAD and CD is the root cause of the tool used in LFVAD to avoid severe edge chipping.

Stabilizability of Bimodal Piecewise Linear Systems With Continuous Vector Field

a parametric model for a cutting body blade cutting edge contour of geokhod, which is a new class of mining equipment. The research relevance is due to the need to create a unified methodology for forming the cutting body cutting edge for geokhods of various sizes, since the cutting edge design parameters are heavily dependent on geokhod geometry and force parameters. The research objective is to obtain parametric equations for determining the geometric coordinates of the points of the geokhod cutting body blade cutting edge in Cartesian coordinates. Methods of research. The paper describes the principles of parameterization of the complex geometry of geokhod cutting body blade cutting edge. Research methods include the study of blade edge geometry, as well as its design parameters dependence on geokhod’s force and design parameters. To achieve that, analytical dependencies were selected from the method of calculating the cutting body force parameters, and then parametric functions affecting the cutting body cutting edge geometry were selected. Results. As a result, a geometric model of the cutting body cutting edge in a parametric form was obtained. To demonstrate the formation of the blade cutting edge contact with rock based on the obtained parametric-time dependencies, a computer program was developed with the calculated values probability construction. The dependence between the cutting edge geometry and the geokhod design parameters dictated the modeling of the cutting edge shape for geokhods of different diameters. Conclusions. The obtained parametric expressions for calculating the coordinates of geokhod cutting body contact make it possible to build contact lines of the blade and the rock mass according to the input data. Is has also been proved that geokhod cutting edge shape depends on the geokhod diameter, and its geometric parameters will be individual for each type of geokhod

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant.

A new radar echo tracking algorithm known as multi-scale tracking radar echoes by cross-correlation (MTREC) was developed in this study to analyze movements of radar echoes at different spatial scales. Movement of radar echoes, particularly associated with convective storms, exhibits different characteristics at various spatial scales as a result of complex interactions among meteorological systems leading to the formation of convective storms. For the null echo region, the usual correlation technique produces zero or a very small magnitude of motion vectors. To mitigate these constraints, MTREC uses the tracking radar echoes by correlation (TREC) technique with a large “box” to determine the systematic movement driven by steering wind, and MTREC applies the TREC technique with a small “box” to estimate small-scale internal motion vectors. Eventually, the MTREC vectors are obtained by synthesizing the systematic motion and the small-scale internal motion. Performance of the MTREC technique was compared with TREC technique using case studies: the Khanun typhoon on 11 September 2005 observed by Wenzhou radar and a squall-line system on 23 June 2011 detected by Beijing radar. The results demonstrate that more spatially smoothed and continuous vector fields can be generated by the MTREC technique, which leads to improvements in tracking the entire radar reflectivity pattern. The new multi-scale tracking scheme was applied to study its impact on the performance of quantitative precipitation nowcasting. The location and intensity of heavy precipitation at a 1-h lead time was more consistent with quantitative precipitation estimates using radar and rain gauges.

Roundabout capacity is primarily estimated by gap-acceptance or by geometric models. The 2010 Highway Capacity Manual (HCM 10) uses a gap-acceptance model developed by Siegloch with empirically derived values of critical gap and headway for single-lane and two-lane entries. A geometric capacity model was developed by Kimber and Hollis that diverged from gap-based models. The capacity of up to four lane entries was empirically derived from six geometric parameters. In 2012 capacity data were collected as part of the FHWA project, Assessment of Roundabout Capacity Models for the Highway Capacity Manual. These data were used to produce HCM6, an update of the HCM 2010. In HCM6, a capacity curve was fitted through all the capacity data for single-lane roundabouts. The large scatter of data about the mean capacity line suggests that the single-lane roundabouts may be separated into different geometric types to improve accuracy. To investigate this hypothesis, the capacity data were separated into two geometrically distinct types of single-lane roundabouts: smaller, compact and larger, curvilinear. The range of data for the disaggregated and aggregated data was compared. Also, a capacity line was derived for each type using the geometric capacity model and compared with the disaggregated data with the HCM6 method. The results demonstrate that differences in geometry, absent in HCM6, explain the wide data range. This was further confirmed by the geometric model that gave a good fit to both sets of data. These results indicate that the accuracy of capacity prediction is improved by including geometric variation.

Several recent works have combined discrete and continuous motion planning methods for robot navigation and control. The basic idea of some of these works is to plan a path, by determining a sequence of neighboring discrete regions of the configuration space, and to assign a vector field that drives the robots through these regions. This paper addresses the problem of efficiently computing vector fields over a sequence of consecutive triangles. Differently from previous numerical approaches, which were not able to compute fully continuous fields in triangulated spaces, this paper presents an algorithm that is able to compute guaranteed continuous vector fields over a sequence of adjacent triangles.

A series of high-speed milling tests of Inconel 718 were carried out utilizing SiAlON ceramic tools, and the transitions of the cutting edge geometry and cutting forces were investigated. Through the experimental investigations, it was confirmed that the cutting edge is worn rapidly and a round shape is formed at the initial stage of machining. The radius of the round cutting edge becomes considerably large with respect to the uncut chip thickness, and thus the ploughing process is dominant in ceramic milling like general micro cutting operations. Based on the observed phenomena, a quasi-mechanistic model for cutting force prediction was proposed, where the measured cutting edge geometry and the contact stress distribution at the toolworkpiece interface are taken into account. The estimated cutting force by the proposed model showed a good agreement with the measured one. Minimizing the estimation error in the cutting forces, contact stresses of the cutting edge to the workpiece are identified. Stress field analysis using the estimated contact stresses revealed that the large tensile stress instantaneously generates around the stagnation point. This mechanism may contribute to the generation of the rake face flaking, which determines the end of the tool life.

We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

Machining process productivity is adversely impacted by the unstable chatter vibrations causing poor surface quality, excessive loads, and premature tool and machine failures. Depending on the position of the cutting edge element (CEE) along the vibration wave on the part surface, its flank face and hone radius dynamically indent into the material and cause process damping (PD), which improves the machining stability. This article proposes a generalized analytical PD model by lifting the straight flank face limitation of authors’ previously developed approach and extends it to any clearance face geometries used in machining operations. The proposed model employs the dimensionality reduction method by discretizing the two-dimensional (2D) contact between the CEE and the part surface with series of springs to simulate the contact mechanics between the two. Elasto-plastic material model of the workpiece is used to calculate the contact pressure considering the work material properties, cutting edge geometry, machining and vibration parameters as inputs. The PD force is evaluated by removing the effect of the static indentation from the overall contact force. The equivalent viscous damping coefficient is calculated to linearize the PD force and can be used for accurate machining stability prediction of difficult-to-cut materials. The model has been validated with the experimentally identified and finite elements-based PD coefficients, and a stability lobes diagram in orthogonal machining. Initial results reveal that the newly proposed analytical model can eliminate the empirical values identified from time-consuming machining tests and computationally expensive numerical simulations. Next, the introduced method will be extended to PD in three-dimensional (3D) machining processes such as milling with general tool geometries having varying cutting speeds and geometries along the tool axis.

A function f defined on [0, 1]^{d} is called strongly chargeable if there is a continuous vector field v such that f(x_{1}, \dots, x_{d}) equals the flux of v through the rectangle [0, x_{1}] \times \cdots \times [0, x_{d}] for all (x_{1}, \dots, x_{d}) \in [0, 1]^{d} . In other words, f is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with d \geq 2 parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter (H_{1}, \dots, H_{d}) are shown to be almost surely strongly chargeable whenever H_{1} + \cdots + H_{d} > d-1 .

Generic flows generated by continuous vector fields in Banach spaces

When Brouwer continued his investigations into Hilbert 5, he discovered that his main topology source, Schoenflies Bericht, was far from correct. He set himself to straighten out the defective parts; the best known fall out of this research was his work on indecomposable continua, with the spectacular example of three domains with one common boundary. The chapter also contains the story of Brouwer’s research on fixed points on the sphere and his translation theorem (on fixed point free continuous maps of the plane onto itself). He simultaneously produced a number of papers on vector field on surfaces. The best known result was the hairy ball theorem: a continuous vector field on a sphere must be zero or infinite at at least one point.